I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges:
$\sum_{\text{known primes}}\frac{1}{p} < 4$.
However these notes were written pre-$2003$. I was wondering if this is still the case. If not then how much bigger has this sum got?
Given the growth rate of $\exp \exp x$, I'd say we should be more careful about the contribution of the $O(1)$ term which is about $0.2615$. That means we only need to know primes up to about $1.8 \times 10^{18}$ in order to reach $4$. Numbers in this range are easily sievable to determine primality — you could essentially compute them as fast as you can record them. So the main bottleneck would be storage: let's assume you require a record of all the primes found, rather than just the reciprocal sum (which would be much, much slimmer).
The total amount of storage needed to hold that many primes is about $2.6 \times 10^{18}$ bits, or roughly 320 petabytes. The number of iPhones sold in the last year is about 160 million, so if each of them was allocated a different range of primes, each would only need to hold 2GB of primes, which I doubt would take more than an hour or two of computation.
So basically, depending on how generous you want to be with the definition of "known prime", Apple could break the $4$-barrier with very little effort in the next iOS system update, if they happened to be so inclined :).
Even if you are less generous and require the list of primes to be held in one location, it is physically possible with current high-capacity tape libraries. Apparently an Oracle StorageTek SL8500 can hold 857 PB in a single large enclosure. Just need to find someone who has a spare one lying around with the tapes to fill it...